Sequence of smooth functions approximating a 2d cylinder step function

Question

Let $f(x,y)=1$ if $(x,y)$ is in the unit disk and $f=0$ otherwise. I would like to approximate $f$ by a sequence of smooth functions. The functions need to be evaluated quickly so the results I'm getting from mollification are to awkward to deal with...

Thanks in advance..

Wait... why did you remove the specific function? o_O – J. M. ain't a mathematician Oct 26 '10 at 07:04 — J. M. ain't a mathematician, Oct 26 '10 at 07:04
Oh whoops... I was editing... that was a mistake... – alext87 Oct 26 '10 at 07:05 — alext87, Oct 26 '10 at 07:05

score 4 · Accepted Answer · answered Oct 26 '10 at 06:53

4

You might want to try any of formulae 17 to 27 in the MathWorld page for the unit step function, since the function you're interested in is expressible as

$$1-H(x^2+y^2-1)$$

answered Oct 26 '10 at 06:53

J. M. ain't a mathematician

75,051

score 1 · Answer 2 · edited Jul 22 '12 at 23:10

1

You can convolve your function with a bivariate Gaussian density kernel, $\Phi(x,y; \sigma_x, \sigma_y)$, and obtain a smooth representation. Then you can let the variance parameters $\sigma_x$ and $\sigma_y$ go to zero to obtain arbitrarily accurate approximation to $f(x,y)$.

edited Jul 22 '12 at 23:10

Henry T. Horton

18,318
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answered Jul 22 '12 at 17:07

Ravi

51

Sequence of smooth functions approximating a 2d cylinder step function

2 Answers2